Differential Mode Structure of the B=1 Skyrmion on S³: Emergent Quarks, Gluons, and Confinement

We show that the B=1 Skyrmion on S³, realised as the identity map U: S³ → SU(2), possesses a natural equatorial decomposition into counter-rotating hemispheres whose differential modes reproduce the quark–gluon structure of the proton with zero free parameters. Key results (all zero free parameters): - Three confined modes (quarks) from l=1 spherical harmonics on the equatorial S²- Eight interaction channels (gluons) from the tensor product 3⊗3 = 1⊕8- Colour neutrality from the cubic selection rule m₁+m₂+m₃=0- Axial coupling g_A = 7/6 = 1.167 (experiment: 1.276, 8.6% — a factor-of-two improvement over the standard Skyrmion value of 0.61)- Quark spin fraction ΔΣ = 1/4 = 0.250 (experiment: 0.28 ± 0.05, 0.6σ — resolves the proton spin crisis)- Constituent quark mass m_const ≈ 347 MeV from relativistic confinement (experiment: ~338 MeV, 2.6%)- Neutron charge radius sign ⟨r²⟩_n < 0 (experiment: −0.116 fm², correct sign)- Neutron-proton mass difference sign m_n − m_p > 0 (experiment: +1.29 MeV, correct sign) The framework provides the explicit Cheshire Cat dictionary between topological soliton structure and partonic degrees of freedom, mapping l=1 torsional modes to quarks, their traceless tensor product to gluons, and the 3j-symbol selection rule to colour confinement. Connection to the fine structure constant: The proven quantities E_cl = 12π² (classical Skyrmion energy, Bogomolny-saturated) and Λ_red = π/8 (reduced torsional moment of inertia) compose the closure parameter of a companion paper's balance equation for the fine structure constant: D = (2E_cl − Λ_red)/Vol(RP³) = 24 − 1/(8π). This exact algebraic identity establishes that the electromagnetic coupling α and the soliton structure share a single geometric substrate. Known limitations are stated explicitly: the Δ–N splitting is 20× too low (structural feature of S³ topology, not a bug), the magnetic moment ratio μ_p/μ_n = −1 vs experimental −1.460, and the N_c derivation assumes the SU(2) hedgehog structure. Standard remedies (variable S³ radius, vector mesons, sixth-order Skyrme terms) would improve dynamics without affecting the differential predictions. The paper is a companion to "Geometric Origin of the Fine Structure Constant" (see Related identifiers).